The Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra over $\mathbb{C}$, let $g$ be its dual Coxeter number, and let $A=\wedge(\mathfrak{g}\oplus\mathfrak{g})/I$, where $I$ is the ideal generated by the three canonical copies of $\mathfrak{g}$ in the bidegree components $(2,0)$, $(1,1)$, and $(0,2)$. Let $A^{\mathfrak{g}}$ denote the invariant subalgebra, and let $S$ be its unique invariant element of bidegree $(1,1)$, namely $S=\operatorname{Tr}|_V(XY)$ for a non-trivial irreducible finite-dimensional representation $V$ of $\mathfrak{g}$. Cachazo-Douglas-Seiberg-Witten conjecture. (i) The algebra $A^{\mathfrak{g}}$ is generated by $S$. (ii) $S^g=0$. (iii) $S^{g-1}\ne 0$. The conjecture concerns the invariant algebra associated with the odd superscheme cut out by the vanishing supercommutators of two copies of $\mathfrak{g}$. It was proposed by Cachazo, Douglas, Seiberg, and Witten and had been proved for classical simple Lie algebras; the source paper proves it for type $G_2$ and verifies a consequence in the general case.